From CFD-Wiki
function [P,Px,Py] = GetPres3W(eu,NodXY,Elcon,nn2nft,Q,EBCPr,~)
% GetPres3W - Compute continuous simple-cubic pressure from velocity
% field on general quadrilateral grid (bilinear geometric mapping) or
% quadratic pressure for triangular grid (linear mapping).
% This version is restricted to 3-node triangles and 4-node quadrilaterals
% P,Px,Py must be reshaped or restructured for use in calling program with
% P=reshape(P,NumNx,NumNy), etc. because it assumes that the grid may be
% unstructured.
%
% Usage
% P = GetPres3W(eu,NodXY,Elcon,nn2nft,Q,EBCPr);
% [P,Px,Py] = GetPres3W(eu,NodXY,Elcon,nn2nft,Q,EBCPr);
%
% Inputs
% eu - velocity class, (eg. ELS3412r, ELS4424r, ELS5424r, ELS2309t )
% NodXY - coordinates of nodes
% Elcon - element connectivity, nodes in element
% nn2nft - global number and type of (non-pressure) DOF at each node
% Q - array of DOFs for velocity elements
% EBCp - essential pressure boundary condition structure
% EBCp.nodn - node number of fixed pressure node
% EBCp.val - value of pressure
% Outputs
% P,Px,Py - pressure degrees of freedom
% Uses
% ilu - ilu preconditioner
% gmres - to solve the system
% Indirectly may use (handle passed by eu):
% GQuad2 - function providing 2D rectangle quadrature rules.
% TQuad2 - function providing 2D triangle quadrature rules.
% ELG3412r - basis function class defining the cubic/pressure elements(Q)
% ELG2309t - basis function class defining the quadratic/pressure elements(T)
%
% Jonas Holdeman, January 2007, revised March 2013
% ------------------- Constants and fixed data ---------------------------
nvn = eu.nnodes; % Number of nodes in elements (4)
nvd = eu.nndofs; % number of velocity DOFs at nodes (3|4|6);
if nvn==4, ep = ELG3412r; % simple cubic voricity class definition (Q)
elseif nvn==3, ep=ELG2309t; % quadratic voricity class definition (T)
else error(['pressure: ' num2str(nvn) ' nodes not supported']);
end
npd = ep.nndofs; % Number DOFs in pressure fns (3, simple cubic)
ND=1:nvd; % Number DOFs in velocity fns (bicubic-derived)
NumEl=size(Elcon,1); % Number of elements
NumNod=size(NodXY,1); % Number of global nodes
NumPdof=npd*NumNod; % Max number pressure DOFs
% Parameters for GMRES solver
GM.Tol=1.e-11;
GM.MIter=30;
GM.MRstrt=8;
% parameters for ilu preconditioner
% Decrease su.droptol if ilu preconditioner fails
su.type='ilutp';
su.droptol=1.e-5;
nn2pft = zeros(NumNod,2);
for n=1:NumNod
nn2pft(n,:)=[(n-1)*npd+1,1];
end
% ---------------------- end fixed data ----------------------------------
% Begin essential boundary conditions, allocate space
EBCp.Mxdof=NumPdof;
% Essential boundary condition for pressure
EBCp.dof = nn2pft(EBCPr.nodn(1),1); % Degree-of-freedom index
EBCp.val = EBCPr.val; % Pressure Dof value
% partion out essential (Dirichlet) dofs
p_vec = (1:EBCp.Mxdof)'; % List of all dofs
EBCp.p_vec_undo = zeros(1,EBCp.Mxdof);
% form a list of non-diri dofs
EBCp.ndro = p_vec(~ismember(p_vec, EBCp.dof)); % list of non-diri dofs
% calculate p_vec_undo to restore Q to the original dof ordering
EBCp.p_vec_undo([EBCp.ndro;EBCp.dof]) = (1:EBCp.Mxdof); %p_vec';
% Allocate space for pressure matrix, velocity data
pMat = spalloc(NumPdof,NumPdof,36*NumPdof); % allocate P mass matrix
Vdata = zeros(NumPdof,1); % allocate velocity data
Vdof = zeros(nvd,nvn); % Nodal velocity DOFs
Xe = zeros(2,nvn);
% BEGIN GLOBAL MATRIX ASSEMBLY
for ne=1:NumEl
Xe(1:2,1:nvn)=NodXY(Elcon(ne,1:nvn),1:2)';
% Get stream function and velocities
for n=1:nvn
Vdof(ND,n)=Q((nn2nft(Elcon(ne,n),1)-1)+ND); % Loop over local nodes
end
% Pressure "mass" matrix
[Emat,Rndx,Cndx] = pMassMat(Xe,Elcon(ne,:),nn2pft,ep);
pMat=pMat+sparse(Rndx,Cndx,Emat,NumPdof,NumPdof); % Global mass assembly
% Convective data term
[CDat,RRndx] = PCDat(Xe,Elcon(ne,:),nn2pft,Vdof,ep,eu);
Vdata(RRndx) = Vdata(RRndx)-CDat(:);
end; % Loop ne
% END GLOBAL MATRIX ASSEMBLY
Vdatap=Vdata(EBCp.ndro)-pMat(EBCp.ndro,EBCp.dof)*EBCp.val;
pMatr=pMat(EBCp.ndro,EBCp.ndro);
%Qs=Qp(EBCp.ndro); % Non-Dirichlet (active) dofs
[Lm,Um] = ilu(pMatr,su); % incomplete LU
Pr = gmres(pMatr,Vdatap,GM.MIter,GM.Tol,GM.MRstrt,Lm,Um,[]); % GMRES
Qp=[Pr;EBCp.val]; % Augment active dofs with esential (Dirichlet) dofs
Qp=Qp(EBCp.p_vec_undo); % Restore natural order
if (nargout==3)
Qp=reshape(Qp,npd,NumNod);
P = Qp(1,:);
Px = Qp(2,:);
Py = Qp(3,:);
else
P = Qp;
end
return;
% >>>>>>>>>>>>> End pressure recovery <<<<<<<<<<<<<
% ********************* function pMassMat ********************************
function [MM,Rndx,Cndx]=pMassMat(Xe,Elcon,nn2nftp,ep)
% Simple cubic gradient element "mass" matrix
% ep = handle to class of definitions for cubic pressure functions:
% ELG3412r (rectangle) or ELG2309t (triangle)
%
% -------------------- Constants and fixed data --------------------------
npn = ep.nnodes; % number of velocity/vorticity element nodes (4)
npd = ep.nndofs; % number of vorticity DOFs at nodes (3);
nn = ep.nn; % defines local nodal order [-1 -1; 1 -1; 1 1; -1 1]
npdf=npn*npd;
NP=1:npd;
% ------------------------------------------------------------------------
persistent QQ_prMMp; % quadrature rules
if isempty(QQ_prMMp)
QRord = (2*ep.mxpowr+1); % quadtature rule order
[QQ_prMMp.xa, QQ_prMMp.ya, QQ_prMMp.wt, QQ_prMMp.nq]=ep.hQuad(QRord);
end % if isempty...
xa = QQ_prMMp.xa; ya = QQ_prMMp.ya; wt = QQ_prMMp.wt; Nq = QQ_prMMp.nq;
% ------------------------------------------------------------------------
persistent ZZ_Gpmm; % pressure functions
if (isempty(ZZ_Gpmm)||size(ZZ_Gpmm,2)~=Nq)
% Evaluate and save/cache the set of shape functions at quadrature pts.
ZZ_Gpmm=cell(npn,Nq);
for k=1:Nq
for m=1:npn
ZZ_Gpmm{m,k}=ep.G(nn(m,:),xa(k),ya(k));
end
end
end % if(isempty(*))
% ------------------------ end fixed data --------------------------------
TGi=cell(npn);
for m=1:npn % Loop over corner nodes, GBL is gradient of bilinear fn
J=(Xe*ep.Gm(nn(:,:),nn(m,1),nn(m,2)))'; %
TGi{m} = blkdiag(1,J);
end % Loop m
MM=zeros(npdf,npdf); G=zeros(2,npdf); % Preallocate arrays
for k=1:Nq
% Initialize functions and derivatives at the quadrature point (xa,ya).
J=(Xe*ep.Gm(nn(:,:),xa(k),ya(k)))'; % Jacobian at (xa,ya)
Det=J(1,1)*J(2,2)-J(1,2)*J(2,1); % Determinant of Jt & J
Ji=[J(2,2),-J(1,2); -J(2,1),J(1,1)]/Det;
for m=1:npn
mm=(m-1)*npd;
G(:,mm+NP)=Ji*ZZ_Gpmm{m,k}*TGi{m};
end % loop m
MM = MM + G'*G*(wt(k)*Det);
end % end loop k (quadrature pts)
gf=zeros(npdf,1); % array of global freedoms
for n=1:npn % Loop over element nodes
m=(n-1)*npd;
gf(m+NP)=(nn2nftp(Elcon(n),1)-1)+NP; % Get global freedoms
end
Rndx=repmat(gf,1,npdf); % Row indices
Cndx=Rndx'; % Column indices
MM = reshape(MM,1,npdf*npdf);
Rndx=reshape(Rndx,1,npdf*npdf);
Cndx=reshape(Cndx,1,npdf*npdf);
return;
% *********************** funnction PCDat ******************************
function [PC,Rndx]=PCDat(Xe,Elcon,nn2nftp,Vdof,ep,eu)
% Evaluate convective force data
% ep = handle to class of definitions for cubic pressure functions:
% ELG3412r (rectangle) or ELG2309t (triangle)
%
% ----------- Constants and fixed data ---------------
nvn = eu.nnodes; % number of velocity element nodes (4)
nvd = eu.nndofs; % number of velocity DOFs at nodes (3|4|6);
npd = ep.nndofs; % number of vorticity DOFs at nodes (3);
nn = eu.nn; % defines local nodal order [-1 -1; 1 -1; 1 1; -1 1]
npdf=nvn*npd;
nvdf=nvn*nvd;
NP=1:npd;
NV=1:nvd;
% ------------------------------------------------------------------------
persistent QQ_prPCp; % quadrature rules
if isempty(QQ_prPCp)
QRord = (eu.mxpowr+ep.mxpowr-1); % quadtature rule order
[QQ_prPCp.xa, QQ_prPCp.ya, QQ_prPCp.wt, QQ_prPCp.nq]=eu.hQuad(QRord);
end % if isempty...
xa = QQ_prPCp.xa; ya = QQ_prPCp.ya; wt = QQ_prPCp.wt; Nq = QQ_prPCp.nq;
% ------------------------------------------------------------------------
persistent ZZ_Spcd; persistent ZZ_SXpcd; persistent ZZ_SYpcd;
persistent ZZ_Gpcd;
if (isempty(ZZ_Spcd)||isempty(ZZ_Gpcd)||size(ZZ_Spcd,2)~=Nq)
% Evaluate and save/cache the set of shape functions at quadrature pts.
ZZ_Spcd=cell(nvn,Nq); ZZ_SXpcd=cell(nvn,Nq);
ZZ_SYpcd=cell(nvn,Nq); ZZ_Gpcd=cell(nvn,Nq);
for k=1:Nq
for m=1:nvn
ZZ_Spcd{m,k} =eu.S(nn(m,:),xa(k),ya(k));
[ZZ_SXpcd{m,k},ZZ_SYpcd{m,k}]=eu.DS(nn(m,:),xa(k),ya(k));
ZZ_Gpcd{m,k}=ep.G(nn(m,:),xa(k),ya(k));
end % loop m over nodes
end % loop k over Nq
end % if(isempty(*))
% ----------------------- end fixed data ---------------------------------
Ti=cell(nvn); TGi=cell(nvn);
for m=1:nvn % Loop over corner nodes
Jt=Xe*eu.Gm(nn(:,:),nn(m,1),nn(m,2));
JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)]; % det(J)*inv(J)
if nvd==3,
TT=blkdiag(1,JtiD);
elseif nvd==4
TT=blkdiag(1,JtiD,(Jt(1,1)*Jt(2,2)+Jt(1,2)*Jt(2,1)) );
else
T2=[Jt(1,1)^2, 2*Jt(1,1)*Jt(2,1), Jt(2,1)^2; ... % alt
Jt(1,1)*Jt(1,2), Jt(1,1)*Jt(2,2)+Jt(1,2)*Jt(2,1), Jt(2,1)*Jt(2,2); ...
Jt(1,2)^2, 2*Jt(1,2)*Jt(2,2), Jt(2,2)^2];
TT=blkdiag(1,JtiD,T2);
Bxy=Xe*eu.DGm(nn(:,:),nn(m,1),nn(m,2)); % Second cross derivatives
TT(5,2)= Bxy(2);
TT(5,3)=-Bxy(1);
end
Ti{m}=TT;
%
% J=[Jt(1,1),Jt(2,1); Jt(1,2),Jt(2,2)]; % evaluate J from transpose
TGi{m} = blkdiag(1,Jt');
end % Loop m over corner nodes
PC=zeros(npdf,1);
S=zeros(2,nvdf); Sx=zeros(2,nvdf); Sy=zeros(2,nvdf); G=zeros(2,npdf);
for k=1:Nq % Loop over quadrature points
Jt=Xe*eu.Gm(nn(:,:),xa(k),ya(k)); % transpose of Jacobian at (xa,ya)
Det=Jt(1,1)*Jt(2,2)-Jt(1,2)*Jt(2,1); % Determinant of Jt & J
Jtd=Jt/Det;
JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)];
Jti=JtiD/Det;
Ji=[Jti(1,1),Jti(2,1); Jti(1,2),Jti(2,2)];
for m=1:nvn % Loop over element nodes
mm=nvd*(m-1);
S(:,mm+NV) =Jtd*ZZ_Spcd{m,k}*Ti{m};
Sx(:,mm+NV)=Jtd*(Jti(1,1)*ZZ_SXpcd{m,k}+Jti(2,1)*ZZ_SYpcd{m,k})*Ti{m};
Sy(:,mm+NV)=Jtd*(Jti(1,2)*ZZ_SXpcd{m,k}+Jti(2,2)*ZZ_SYpcd{m,k})*Ti{m};
mm=npd*(m-1);
G(:,mm+NP)=Ji*ZZ_Gpcd{m,k}*TGi{m};
end % end loop over element nodes
% Compute the fluid velocities at the quadrature point.
U = S*Vdof(:);
Ux = Sx*Vdof(:);
Uy = Sy*Vdof(:);
UgU = U(1)*Ux+U(2)*Uy; % U dot grad U
PC = PC + G'*UgU*(wt(k)*Det);
end % end loop over Nq quadrature points
gf=zeros(1,npdf); % array of global freedoms
for n=1:nvn % Loop over element nodes
m=(n-1)*npd;
gf(m+NP)=(nn2nftp(Elcon(n),1)-1)+NP; % Get global freedoms
end
Rndx=gf;
PC = reshape(PC,1,npdf);
return;
